Answer
$P(x)=(x-5)(x+5)(x+5i)(x-5i)$
The zeros of the function are: $\{5i,-5i, 5, -5\}$
$x =5$ with multiplicity 1
$x =-5$ with multiplicity 1
$x =5i$ with multiplicity 1
$x =-5i$ with multiplicity 1
Work Step by Step
Factor the polynomial completely to obtain:
$P(x)=x^{4}-625$
$P(x)=(x^{2}-25)(x^{2}+25)$
$P(x)=(x-5)(x+5)(x+5i)(x-5i)$
Equate each unique factor to zero then solve each equation to obtain:
$x-5=0 \rightarrow x=5$
$x+5=0 \rightarrow x=-5$
$x+5i=0 \rightarrow x=-5i$
$x-5i=0 \rightarrow x=5i$
The zeros of the function are: $\{5i,-5i, 5, -5\}$
$x =5$ with multiplicity 1
$x =-5$ with multiplicity 1
$x =5i$ with multiplicity 1
$x =-5i$ with multiplicity 1
